Binary Logistic Regression
STA6235: Modeling in Regression
Introduction
We have previously discussed continuous outcomes and the appropriate distributions.
Normal distribution
Gamma distribution
Let’s now consider categorical outcomes:
Binary
Ordinal
Multinomial
Binary Logistic Regression
- We model binary outcomes using logistic regression,
\ln \left( \frac{\pi}{1-\pi} \right) = \beta_0 + \beta_1 x_1 + ... + \beta_k x_k,
where \pi = \text{P}[Y = 1] = the probability of the outcome/event.
How is this different from linear regression? y = \beta_0 + \beta_1 x_1 + ... + \beta_k x_k
or Gamma regression?
\ln(y) = \beta_0 + \beta_1 x_1 + ... + \beta_k x_k
R Syntax
Today’s Data
- Today we will be using the Roy Kent dataset from Tidy Tuesday.
library(tidyverse)
richmondway <- read_csv('https://raw.githubusercontent.com/rfordatascience/tidytuesday/master/data/2023/2023-09-26/richmondway.csv') %>%
mutate(dating = if_else(Dating_flag == "Yes", 1, 0),
IMDB = if_else(Imdb_rating >= 8.5, 1, 0)) %>%
select(Season, Episode, F_count_RK, F_perc, dating, IMDB)
# quantile(richmondway$Imdb_rating, c(0, 0.25, 0.5, 0.75, 1))
# richmondway %>% count(IMDB_9) Example: Roy Kent’s F-Bombs
- Let’s model the odds of Roy Kent and Keeley Jones dating in a particular episode (dating) as a function of the percentage of F-bombs that belong to Roy Kent (F_perc) and if the IMDB rating is an 8.5 or better (IMDB).
m1 <- glm(dating ~ F_perc + IMDB,
data = richmondway,
family = "binomial"(link="logit"))
summary(m1)
Call:
glm(formula = dating ~ F_perc + IMDB, family = binomial(link = "logit"),
data = richmondway)
Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept) -1.76166 1.08995 -1.616 0.106
F_perc 0.03323 0.02506 1.326 0.185
IMDB 0.37986 0.72250 0.526 0.599
(Dispersion parameter for binomial family taken to be 1)
Null deviance: 46.662 on 33 degrees of freedom
Residual deviance: 44.261 on 31 degrees of freedom
AIC: 50.261
Number of Fisher Scoring iterations: 4Example: Roy Kent’s F-Bombs
The model is as follows,\ln \left( \frac{\hat{\pi}}{1-\hat{\pi}} \right) = -1.76 + 0.03 x_1 + 0.38 x_2, where
x_1 is the episode’s percentage of the F-bombs from Roy Kent
x_2 is the IMDB rating categorization of the episode
- 0 = IMDB rating < 8.5
- 1 = IMDB rating \ge 8.5
Odds Ratios
Recall the binary logistic regression model, \ln \left( \frac{\pi}{1-\pi} \right) = \beta_0 + \beta_1 x_1 + ... + \beta_k x_k,
We are modeling the log odds, which are not intuitive with interpretations.
To be able to discuss the odds, we will “undo” the natural log by exponentiation.
i.e., if we want to interpret the slope for x_i, we will look at e^{\hat{\beta}_i}.
When interpreting \hat{\beta}_i, it is an additive effect on the log odds.
When interpreting e^{\hat{\beta}_i}, it is a multiplicative effect on the odds.
Odds Ratios
- Why is it a multiplicative effect?
\begin{align*} \ln \left( \frac{\pi}{1-\pi} \right) &= \beta_0 + \beta_1 x_1 + ... + \beta_k x_k \\ \exp\left\{ \ln \left( \frac{\pi}{1-\pi} \right) \right\} &= \exp\left\{ \beta_0 + \beta_1 x_1 + ... + \beta_k x_k \right\} \\ \frac{\pi}{1-\pi} &= e^{\beta_0} e^{\beta_1 x_1} \cdots e^{\beta_k x_k} \end{align*}
Odds Ratios
For continuous predictors:
- For a 1 [unit of predictor] increase in [predictor name], the odds of [outcome] are multiplied by [e^{\hat{\beta}_i}].
- For a 1 [unit of predictor] increase in [predictor name], the odds of [outcome] are [increased or decreased] by [100(e^{\hat{\beta}_i}-1)% or 100(1-e^{\hat{\beta}_i})%].
For categorical predictors:
- As compared to [the reference group], the odds of [outcome] for [group that the slope belongs to] are multiplied by [e^{\hat{\beta}_i}].
- As compared to [the reference group], the odds of [outcome] for [group that the slope belongs to] are [increased or decreased] by [100(e^{\hat{\beta}_i}-1)% or 100(1-e^{\hat{\beta}_i})%].
Example: Roy Kent’s F-Bombs
Let’s interpret the odds ratios:
For a 1 percentage point increase in the percentage of f-bombs that came from Roy Kent, the odds of Roy and Keeley dating increase by 3%.
As compared to when episodes have less than an IMDB rating of 8.5, the odds of Roy and Keeley dating are 46% higher in episodes with an IMDB rating of at least 8.5.
Tests for Significant Predictors
- What we’ve learned so far re: significance of predictors holds true with logistic regression
- Looking at the results from
summary():
Call:
glm(formula = dating ~ F_perc + IMDB, family = binomial(link = "logit"),
data = richmondway)
Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept) -1.76166 1.08995 -1.616 0.106
F_perc 0.03323 0.02506 1.326 0.185
IMDB 0.37986 0.72250 0.526 0.599
(Dispersion parameter for binomial family taken to be 1)
Null deviance: 46.662 on 33 degrees of freedom
Residual deviance: 44.261 on 31 degrees of freedom
AIC: 50.261
Number of Fisher Scoring iterations: 4Using the full/reduced Partial F approach:
Note 1! I am showing this for demonstration purposes; we do not need a Partial F for this particular model.
Note 2! We must add
test = "LRT"to theanova()function.
Test for Significant Regression Line
What we’ve learned so far re: significance of predictors holds true with logistic regression
- Significance of regression line \to ANOVA with full/reduced models
Predicted Values for Data Visualization
The guidelines we’ve set up for data visualization still hold true.
We will put our outcome on the y-axis and a continuous (or at least ordinal) predictor on the x-axis.
- We will hold all other predictors constant in order to create lines.
Recall the logistic regression model, \ln \left( \frac{\pi_i}{1-\pi_i} \right) = \beta_0 + \beta_1 x_{1i} + ... + \beta_k x_{ki}
We can solve for the probability, which allows us to predict the probability that y_i=1 given the specified model: \pi_i = \frac{\exp\left\{ \beta_0 + \beta_1 x_{1i} + ... + \beta_k x_{ki} \right\}}{1 + \exp\left\{ \beta_0 + \beta_1 x_{1i} + ... + \beta_k x_{ki} \right\}}
Example: Roy Kent’s F-Bombs
